# Algebra Review: Area of a Trapezoid

There are lots of nice and simple and very visual derivations of the area of a trapezoid. Those are the ones I hope my students hang on to and rely on when they are thinking about how to find and calculate its area. (We’ve already talked about my reservations with simply memorizing area formulas, and the problems that can arise from rote memorization).

However, in my geometry classes, I like to present this algebraic derivation of the area formula for a trapezoid. This is about the point in the year where students have forgotten a lot of their algebra skills from last year, and even cycling through some of those skills in classwork and homework only does so much. Playing around with some ideas, including thinking about a trapezoid as a decapitated triangle (because that’s clearly what it is, right?) led me to this derivation. I won’t claim that I’m the first to come up with it, and searching around online I’ve found some similar derivations, but I really like this one because of all the ideas it touches on: similar triangles, somewhat complicated proportional relationships, distribution and factoring of algebraic expressions, manipulating fractions, and more. Even though a few eyes start to glaze over when I first tell students I’m going to lead them through a derivation filled with lots of algebra, they start to get interested when they see that there are places where those algebra skills can really help make sense of a situation.

Here’s the derivation:

Deriving the Area of a Trapezoid (With Lots of Algebra Review)

Suppose we have trapezoid $\square BCED$ with an altitude of length h drawn through points F on $\overleftrightarrow{BC}$ and G on $\overleftrightarrow{DE}$. Let $BC=b_1$ and let $DE=b_2$.

Extend the legs until they intersect at point A, creating $\triangle ADE$ and $\triangle ABC$. Since $\overline{BC} \parallel \overline{DE}$, we know that $\triangle ABC \thicksim \triangle ADE$. Also, label the intersection of $\overline{DE}$ and the line perpendicular to $\overline{DE}$ through A as G, and label the intersection of $\overline{BC}$ and $\overline{AG}$ as F. Then $FG = h$ is the altitude of trapezoid $\square BCED$, and the altitude of  $\triangle ABC$ is $AF=h_2$.

Since we have similar triangles, we can write the following proportion: $\frac{h_2}{h}=\frac{b_1}{b_2 - b_1}$. Also, since $\alpha \triangle ABC + \alpha \square BCED = \alpha \triangle ADE$, we can subtract to find the area of the trapezoid: $\alpha \square BCED = \alpha \triangle ADE - \alpha \triangle ABC$. The areas of the triangles can be calculated: $\alpha \triangle ABC = \frac{1}{2} (b_1)(h_2)$ and $\alpha \triangle ADE = \frac{1}{2} (b_2)(h_2 + h)$. When we substitute the area calculations, we find that:

$\alpha \square BCED = \frac{1}{2} \big[(b_2)(h_2 + h) - \frac{1}{2} (b_1)(h_2)\big]$

At this point, we have the area of the trapezoid in terms of the measurements of its two bases (b1 and b2), the measurement of its height (h), and the measurement of the height of , h2. We can use the proportion we found, $\frac{h_2}{h}=\frac{b_1}{b_2 - b_1}$, and solve it for h2 to find  $h_2=\frac{b_1 h}{b_2 - b_1}$. Substituting this in the equation above gets an expression for the area of trapezoid  in terms of the two bases and the height of the trapezoid.

$\alpha \square BCED = \frac{b_2 h + b_2 \bigg(\frac{b_1 h}{b_2 - b_1}\bigg) - b_1 \bigg(\frac{b_1 h}{b_2 - b_1}\bigg)}{2}$

Factoring out $\frac{1}{2}$ and simplifying the fractions:

$\alpha \square BCED = \frac{1}{2} \bigg( b_2 h + \frac{b_1 b_2 h}{b_2 - b_1} - \frac{(b_1)^2 h}{b_2 - b_1} \bigg)$

Then multiply $b_2 h$ by $\frac {b_2 - b_1}{b_2 - b_1}$ to make all the terms inside the parentheses have the same denominator:

$\alpha \square BCED = \frac{1}{2} \bigg( b_2 h \bigg( \frac {b_2 - b_1}{b_2 - b_1} \bigg ) + \frac{b_1 b_2 h}{b_2 - b_1} - \frac{(b_1)^2 h}{b_2 - b_1} \bigg)$

Now distribute and place everything inside the parentheses into one fraction and simplify:

$\alpha \square BCED = \frac {1}{2} \bigg( \frac {(b_2)^2h - b_1 b_2 h + b_1 b_2 h - (b_1)^2h}{b_2 - b_1} \bigg)$

$\alpha \square BCED = \frac {1}{2} \bigg( \frac {(b_2)^2h - (b_1)^2h}{b_2 - b_1} \bigg)$

Factoring out the $h$ leaves:

$\alpha \square BCED = \frac {h}{2} \bigg( \frac {(b_2)^2 - (b_1)^2}{b_2 - b_1} \bigg)$

Then factoring the difference of squares gives:

$\alpha \square BCED = \frac {h}{2} \bigg( \frac {(b_2 + b_1)(b_2 - b_1)}{b_2 - b_1} \bigg)$

Now divide out the $b_2 - b_1$ and rearrange to get our familiar area formula:

$\alpha \square BCED = \frac {h (b_1 + b_2)}{2}$

There are a lot of things I like about this approach, but the engagement of students as I lead them through this proof is a lot of fun. I rarely stand up and lecture or give students almost everything in a proof, but there are times when lecturing is effective and shouldn’t be summarily abandoned. Plus, this also gives me a chance, as we go through each step, to walk around and see which students may need more help with fractions or factoring as we continue through the year, and I can give them some additional help and resources. It’s also an opportunity to talk with the Algebra 2 teacher about the upcoming class and their current skill levels with important concepts.

# A Curiousity in Geometry

One thing that I try to emphasize to my students is that mathematicians don’t spend their time (usually) doing calculations. They spend a lot of time trying to make sense of scenarios, and trying to answer bigger questions by breaking them down, and they spend a lot of time being curious. One of the most important skills of a mathematician, to me, is to try to find a good question. Last week, we had one of my favorite geometry lessons of the year, in which we just spend the entire class playing and posing questions.

I asked students to spend the first 10 minutes of class just constructing isosceles triangles, and whatever may go along with them. At this point in the year, we have covered constructions of perpendicular bisectors, midpoints, angle bisectors, and we just discovered a variety of triangle centers: orthocenters, centroids, incenters, and more. Students are encouraged to use either pencil and paper, GeoGebra, or both. After playing around and having several sketches and constructions completed, I encourage them to spend 5 minutes looking at their own drawings as well as those of the other three group members, and do a little noticing and wondering. While they do this, I model this process as well, both with a doc camera and with GeoGebra on my SmartBoard.

I’m always impressed with the variety of questions that students come up with, and this year was no different:

Check out how creative these students are! How curious! How thoughtful! How perceptive! I encouraged them to try to find questions that they don’t know the answers to, and then challenged them to find questions that they think I don’t know the answers to.

I’d also like to share something that I found, when playing around with GeoGebra. I constructed an isosceles triangle based on a circle, and then constructed the orthocenter, and then animated one of the legs as I traced the path of the orthocenter.

I made a video and narrated it, showing the steps as well.

Well, it turns out that when you do interesting things and share them on Twitter, you get a wonderful variety of interesting other things happening.

Mark Kaercher (@shskaercher) had all sorts of brilliant ideas to explore:

Steve Phelps (@giohio) suggested tracing the incenters and excenters together, and later put a whole bunch together:

Henri Piccioto (@hpicciotto) modeled my original question in Cabri:

And then Henri explored further:

I was thrilled to watch this thread, although I barely was able to take part. You see, I started this post on Monday morning before school, and partway through my second class on Monday, a family emergency came up and I’ve been otherwise occupied for most of the time since then. For anyone I haven’t told who reads this and is curious, please feel free to reach out and ask. Everything isn’t great or okay, but I’m being cautiously optimistic at the moment, and really yearning to get back to the classroom and back to the world of math and teaching and a sense of normalcy.

# List of Blog Posts to Write

This year, I was so excited. I have a prep period, I stopped requiring (and grading and commenting on) homework assignments, and suddenly had so much free time and no excuse to not write. Except that sometimes, life happens. This wasn’t a case of anything catastrophic, or even bad. In fact, a lot of it was good. I started attending math teacher circles each month. I spend more time with my wife and daughters at night and on weekends. I am getting more sleep (except when our three year old decides to crawl into our bed at 3AM, kick me, and take my covers). After Carl Oliver’s great blog post, and follow up presentation at Twitter Math Camp to #PushSend, I really wanted to write more regularly, whenever thoughts came into my head. And yet here it is, just before Thanksgiving, and I haven’t written anything since the beginning of the school year.

It’s not like there haven’t been a whole lot of things on my mind. Maybe having a list of things that I wanted to write about will help me make it happen. So here it is…the list of phantom blog posts:

• Thoughts on my math students who have received cultural messages that they are not cut out to do math, whether due to disability, due to ethnicity, due to race, due to gender, due to speed, due to athleticism, or other factors that I may not be aware of, and what I can be doing better.
• Thinking about a quadratic equation as both a product of lines and as a sum of a quadratic, linear, and constant term, and how to think more geometrically about algebra.
• Development of formal debate in my math classes, including the technology that we use to make debates accessible to all.
• That non-binary student who is often mis-gendered, and my uncertainty about whether I have taken the right approach each time.
• Reasons that I teach proof at the beginning of Geometry (which isn’t a novel sequencing, but one that I’ve thought a lot about).
• Experiences with not requiring homework in most cases, including in-class homework reflections, increases in how much homework is done (admittedly through self-reporting from students), and decreases in stress and workload for students and for me.
• Year 2 of my implementation of standards based grading (in a school that uses a traditional grading system on report cards), and what I learned from my first year.
• The amazing things my math club does every week, largely with ideas mined from #MTBoS (Twitter Math Educator Community).
• Working through my implementation of a CPM Algebra 1 curriculum, which I largely like but find it difficult to deviate from.
• The good, the bad, and the ugly of the Barbie Bungee project.
• The number of very capable Pre-Calculus students who decide not to take on the challenge of doing the class at an honors level, and how I can make it more enticing, and not just seem like more work.
• The awesomeness of our faculty band.
• Experiencing Imposter Syndrome in year 17 of teaching.
• Playing with math games in my core (SET, Wits and Wagers, and Prime Climb are favorites so far).

On a good note, I have continued to be relatively active on Twitter, and should (hopefully) be well prepared for my talk in (eek!) 10 days at Asilomar!

# Simplifying with Complexity

Sometimes the quickest and easiest way to solve an easier problem isn’t the best way to approach a hard problem. Today (first day of course content in my Pre-Calculus class) we were reviewing sets of numbers, categories of numbers, and notation. I gave my class the following set:

{-2, -1, 0, 1, 2, 3}

and asked them to turn it into set builder notation. Every student was able to write the set as some variant of the following:

I asked if there were other ways of writing it, and some people changed the set of integers to whole numbers or natural numbers, but that was about it.

Then I asked the class to do the same thing with the following set:

{3, 6, 9, 12 … }

And they struggled, and struggled. I saw a lot of students just going nowhere. As they worked, I went back to that original set, and wrote up a more complicated solution:

One student looked up, then another. A couple of students started working out how my solution worked. A couple of others asked for clarification. And then, one by one, I started to hear those magical “Ohhh!” sounds that we all love – you know that light bulb moment. Students started to remember that they could match each term to a number, and most of them came up with something like this:

From a desk off to the side, I heard a student exclaim, “Whoah! When you showed us a more complicated way to do the easy problem, it made it easy to do the hard problems!”

# Day 1 Debate – What’s the Best Number?

I know that by now, most schools have started. I think I’m one of the last that still starts after Labor Day – not that I’m complaining. A popular blog post out there is a first day activity, and I wanted to share the one that I’ve done the past few years. Yes, it’s a debate, but a totally informal debate, one that works at any level of mathematical background, students have a lot of fun with, that builds both competition and teamwork in a low stress way, and tells me a surprising amount about my students.

After normal introductions, I give students a simple task – to come up with a number. It could be a favorite number, or a really interesting number, or a number that has some personal meaning. I then ask them, once they’ve decided on a number, to come up with as many interesting things about that number as they can, and give them a couple of minutes. They can use calculators, they can use the Internet, they can draw, and if they get stuck they can ask me for help (though, to be honest, they rarely ask for help with this).

After a few minutes, when at least some of the students are feeling like they’re done, I have them get into pairs, and then in the pair they decide which number is better. They are given about one minute to make their cases and decide, and once each pair has decided, I have each pair find another pair, and decide which of the two numbers is better. It’s interesting to me that I never actually describe this to students as a debate, and in theory they are working together, but they do have something invested in the number that they came with. Inevitably, groups start to argue, but generally nicely, and the whole idea of comparing the best and worst qualities of numbers becomes a source of passion.

The process of finding another group and then discussing, then finding another group and discussing, continues until you have (hopefully) two halves of the class shouting at each other about whether 32 is a better number than 360 (because powers of 2 are more important than having lots of factors and describing the degrees of a circle), or whether 12 is better than 18 (but of course 12 is better).

In listening to conversations that happen, I can get to know an amazing amount about student interests, as well as which students feel very comfortable with what numbers mean and how they can be manipulated and described mathematically. On top of it all, having an entire class passionately engaged in a meaningless debate about which number is best, where you can catch every student having fun playing with math from day one, is a pretty great way to start the year in my opinion.

So, what’s your favorite number? Why?

# Seeking My Role in Diversity in HS Math Classes

I want to be clear. I teach in a private high school in Silicon Valley. Many* of the students at my school are white and come from wealthy families. Most of them have gone to either very good public schools or very good private schools, where most students received a generally good education in mathematics. It may not have been perfect, and our students may have slipped through the cracks, or been told that they weren’t math people, or somehow may have received the message that higher math wasn’t going to be for them. Yet those students were still exposed to the important ideas that they were expected to see, from fractions through basic algebra, from area formulas through linear equations and graphs. When that group of students has “Algebra 1” on their transcript, and they received a B in the class, we have no reservations about putting them into a Geometry class.

Our school also has a large number of students of color, and many* of our black and brown students came from very different schools. While most of Silicon Valley is quite expensive, there are pockets and neighborhoods up and down the peninsula that are considered low income areas. In some of these neighborhoods’ schools, some of our students receive a very different math education. I have seen students who received an A in an Algebra 1 class who had never seen a parabola, who had never factored a trinomial, and who were not consistently able to solve a single variable linear equation. In most cases, this was no fault of their own, and it is not my place to fault their Algebra 1 teacher.

These two different experiences are not an accident. Make no mistake about it, this is systematic racism. As Morgan Fierst posted in a conversation on twitter:

She is absolutely right. But what to do about it? The obvious answer is to dismantle the system, but how does that happen? There is definite harm happening in some elementary and middle schools that serve primarily students of color, but one thing that has become clear to me is that, as a white male high school teacher, I have no right to go in and tell other teachers, especially K-8 teachers of color, how to do their job better. My role is to find the leaders among K-8 teachers and teacher leaders of color and support them, and back them up, and give them my power to dismantle the system.

And what about my school? One of the deciding factors in me taking a job at my school was the high retention and low turnover rate. In my four years, we have had a science teacher retire (and then pass away), an art teacher go to graduate school, and a sign language teacher decide to become a stay at home mom. We have hired three outstanding replacements for those teachers, but only one was a person of color. One third of new faculty hires being non-white is an impressive number if we were hiring 200 people or 40 people, but not when hiring only three. I am not in charge of hiring, and I don’t know how much of an emphasis was made on looking for non-white teachers to interview. We are a small school and don’t have a lot of resources for hiring, and we are not a target school for lots of graduates of teacher credential programs. Maybe we couldn’t have done any better.

Our Head of School retired this past year, and there was an exhaustive search for just the right candidate. Our search committee decided on three very competent finalists, and again, one out of those three was a person of color. My question to each of those candidates was the same: “Our school prides itself on the diversity of our student body, but our faculty doesn’t look the same. We have an amazing and talented group of teachers, but we are mostly very white. Without firing faculty members, how would you improve the diversity of our teachers and staff?” It was an unfair question, and one without an obvious answer, but it was also a question where it was clear which one of the members had given it a lot of thought long before I had asked about it. No surprise, it was Phil Gutierrez, the one candidate who hadn’t lived with white privilege, and I am very happy that he is now on board as our new Head of School. I don’t think he has the answer (because, really, does anyone have the answer yet?), but I do feel that he has the same goal in mind.

For me, in my closed world of math education, the goal is to make sure that the higher level math classes have the same diversity as our general student population, and that our students who choose STEM careers in more rigorous schools are a diverse group of students. However, the end result of those students who enter 9th grade not prepared for  success in Algebra 1 or Geometry (despite what their transcript may say) is that they don’t take higher math or attend rigorous schools or choose STEM careers at the same rate as their white peers. They end up either taking a Pre-Algebra class and end up “behind”, or they struggle to keep up in their Algebra 1 or Geometry class, doing lots of extra work and getting extra help to catch up to their peers on the fly. The extra work and extra help takes extra energy and time that they frankly shouldn’t have to put in. Yet, what options do they have? What options do I have? And what options does our school have?

*To be clear, there is a diversity of economic backgrounds within each ethnic group, and I don’t have the hard data. I believe it is sufficient to say that most of our students from wealthier families are white and many of our students of color come from families and neighborhoods that most would consider lower-income. There are always exceptions to these generalizations. I also acknowledge that I am only discussing white, black, and brown students, and leaving out other significant parts of our population. I also haven’t brought learning differences into this post, which would further complicate the discussion, but these should all be important parts of any discussion of equity in education. I guess that’s the difference between an informal blog post and the book I wish I had the time (and skill) to write.

# Functions – Operations, Transformations, Compositions

Several years ago, I taught PreCalculus from the COMAP PreCalculus: Modeling Our World (1st edition), which was a textbook that I really appreciated. It was very focused on good applied problems, on building conceptual understanding, and on avoiding lots of drill and kill style problems so prevalent in so many textbooks. I still use some of its problems as sources in my classes, but I did find that its lack of clear structure to its units, as well as minimal specific “vocabulary/theorems/algorithms to learn/memorize” was quite unpopular with students.

One of my favorite parts of the text was that it developed the idea of functions as a set of tools for modeling data. Based on the data that you are given, you determine which tool may be your best option. This led to a natural desire to transform or combine functions to make more sophisticated models. Suddenly, we could look at a polynomial in two different ways – is it a product of linear equations, or is it a sum of power functions? Depending on the situation being modeled, maybe one approach makes more sense than the second. And what happens if we want to divide one function by another? Suddenly, we can end up with a rational function, which can drastically change our end behavior and get us talking about a limit. What if we want to sum up different sinusoidal functions to approximate graphs that we see on an oscilloscope? And voila, we are exploring Fourier series!

The great part of thinking about a toolkit of parent functions and the various compositions, operations, and transformations on those functions, is that it allows a student to generalize what happens for any function, be it a direct variation, a sine function, a log function, or other. Playing around with Desmos makes these connections so much easier to see!

# Inclusiveness in Math Education (#TMC17 Theme?)

Twitter Math Camp (TMC17) is over for this year. It took two days to start this post, and over two weeks to finish it, and there is still so much to process. This was my first one, and I’m sure that some parts are always the same, but other parts are surely unique to this year. If I had to pinpoint one overarching theme for the last week, though, it wasn’t directly about math at all.

Make no mistake – I did a lot of math, and had a lot of fun learning about new problems, playing around with new ideas, and discovering new mathematicians who I hope will continue to teach me such interesting bits of mathematics. But that wasn’t the most important part of the experience. If I were to sum up the most important part of the week in one word, it would be inclusiveness.

When I first arrived in Atlanta a week ago, I got the opportunity to meet up with an old friend, Shebah. I taught with her in Oakland almost ten years ago, and she has long since moved across the country. She comes from a family of Mexican heritage, and is engaged to a man with a Puerto Rican background. As we talked about my career, and my colleagues in math education, and this whole community of Twitter teachers, she asked about the diversity of TMC. I only knew a lot of people from their profile pictures, and although I can point out some people of color, that just highlights the lack of racial or ethnic diversity. I did mention that there is a lot of gender diversity in terms of men and women, although I do not know (nor is it really my business unless a friend or colleague chooses to share with me) how many identify as trans/non-binary/genderqueer. Is there diversity of sexual orientation? My experience is that gaydar is not to be trusted, so except for people who mentioned the gender of their partner or spouse, peoples’ sexual orientation just didn’t come up. So how diverse is TMC? The only answer I can say with confidence is that it’s not diverse enough. And right away, I had the idea of inclusion in the back of my mind. Do people who are not white feel included in the Math Twitter Blog o Sphere?

Wednesday was the Desmos Pre-Conference day, a day in which I got to see some amazing new developments in store for Desmos users, including more control for scripting when writing activities, enhancements to the Desmos Geometry app, and some interesting transformations to play with.  There was a great evening activity put on by Desmos, and I went to sleep that night so excited to be in my little world of nerdy math teachers.

And then, on Thursday, July 27, Dan Meyer published a blog post, “Let’s Retire #MTBoS”. And as a result of that post, lots of people over the next several days became hurt, angry, and felt disrespected and dismissed. Again the theme of this whole episode boiled down to inclusion. Who feels included in the #MTBoS community? Who doesn’t feel included? What can be done to bridge those gaps, to make every math education professional, new or experienced, K-12 and beyond, coaches and administrators, all feel a part of this community?

Through the rest of the week, themes of inclusion and belong arose – from which teachers felt welcome in #MTBoS (whether due to its perception as cliquish, the relative youthfulness of the organization, or due to the smaller number of non cis-white members), why there was such a relatively small number of elementary school teachers or minority teachers at TMC, to how we can improve the status and inclusion of students of color and non-male/non-binary students. A whole Twitter thread (or variety of threads) on the topic of what can and should replace “boys and girls” or “ladies and gentlemen” made its way into a Storify Posting on #Equity. Some preferred the idea of using “y’all” or a variation, some liked “scholars”, “learners”, or “my little monsters”, and some defended the older teachers who used the traditional “boys and girls” because it’s hard to change. I don’t buy that argument at all – we don’t (or shouldn’t) accept when teachers from earlier generations maintain their stereotypes and outdated language.

I have memories of my great aunt, a schoolteacher, talking about some of her “colored students” with a surprised affection, like they were overachieving in her eyes because they would sometimes perform at the same level as her “regular” students. I didn’t stand up to her comments at the time, since I was probably about 15 years old, she had been retired for probably 20 years, and my mindset was that it probably didn’t matter too much what she said in the privacy of her own home. Still, I know now that it did matter. I may have silently disagreed with her, but other people who she talked to may have been swayed by her statements. I feel like I have come a long way, but sometimes wonder just how forcefully I would confront a teacher who, whether blatantly or subtly, whether intentionally or accidentally, spoke in a manner that was offensive towards a student or group of students. And then I realize that I have a very mixed record, and that I’ve let teachers slide, not saying something in the moment, because I don’t want to get into a confrontation that will take an important discussion on a tangent. I definitely swallowed my tongue on a few occasions with parents as well when they have said things that offended me greatly. My goal for the future – to take that stand, even when it may be uncomfortable, even when it may cause some unwanted ripples. To allow a message of exclusivity, whether it means excluding teachers from our professional community or excluding students from the class culture, to be voiced without objecting to it is tacitly endorsing something that can’t happen.

So, amazingly, despite all of the great mathematical discussions and ideas that came out of TMC17, which were definitely the most fun, the ideas of equity in the math education community are a far more important takeaway to me. In light of the events in Charlottesville this past weekend, this theme is more important than ever.

# #MTBoS vs #iteachmath Debate

Wow – what an amazing experience #TMC17 was. Including the Desmos pre-conference, it was 4.5 days of cultivating relationships, strengthening friendships, learning from peers, sharing my own experiences, socializing during meals and games, and soaking in a positive experience. And yet…a debate broke out on twitter that was marked by very strong feelings from two different camps. The short story is that Dan Meyer suggested (through tweets and a blog post) that it is time to retire the #MTBoS name and start using the hashtag #iteachmath instead. The response from many in #MTBoS was swift and unrelenting. And some of us mostly stayed on the sidelines, trying to process just what #MTBoS means to us.

I don’t have any answers yet, but I do have some thoughts. First, one thing that I think may have gotten lost is that both sides are really coming at this from a positive place. Maybe it’s naïve of me to believe that, but it’s also important to me that I believe that. It’s not just that I hate conflict (though that’s true), but more that I need to believe that as educators, we are people who care about our kids and our craft.

The #MTBoS does not have any membership application or member fees or anything official. It is (as Peg Cagle said this morning at breakfast) an organization where one joins by participating. Participating could be through writing blogs, through tweeting, through reading blogs, or even by lurking on Twitter.  People who were otherwise introverted and reluctant to reach out in real life at conferences, or who were in small or isolated schools and school districts, found a home in this virtual community. It grew and became a family, a network, a web of relationships. And as it grew, it took on a life of its own. Its members, many of whom identify as introverts, found an avenue to become leaders in math education, and found a community that they could love and call home.

At the beginning of my teaching career, oh, how I wish I had found #MTBoS, but it was several years before Twitter came along, and the only blogging I knew about was on LiveJournal, mostly by people much younger than I was who used it as a personal (but public) diary.  I taught in a small school, was one of only two math teachers at most in the high school, and lapped up the opportunities I had to attend the CMC-North conferences in Asilomar every year, and the NCTM Annual conferences when I could get the funding. But I was mostly in a bubble.  When I heard a few people talk about Twitter and math education almost ten years ago, I just wasn’t sold. I mean – what can be said in 140 characters? How could that be at all meaningful? And who has the time to read and write blogs?

In April 2015, I somehow got around to joining Twitter for real. I knew about a few people, and knew a few others, and eventually found a small group that I felt comfortable following, reading tweets, and reading their blogs. By the time NCTM 2016 came around, I had an idea of what #MTBoS was, and sort of felt like maybe I was on the verge of belonging. To a degree, it seemed like an exclusive group, not intentionally perhaps, but a group that was close and had developed great bonds with each other, and I wondered if I would really fit in.

And so that brings me back to the two sides of the debate. On one hand, there are many, many math educators who either don’t know about #MTBoS, don’t see the value in #MTBoS, or don’t feel invited to participate in #MTBoS. I know that they would of course be encouraged to “join” and would be welcomed with open arms, but they don’t know that. It has been said that #MTBoS wants to increase its diversity, especially in terms of people of color and in terms of more elementary school level teachers. These are necessary goals, and worthwhile goals, and something that we really need to figure out as a community if we want #MTBoS to best serve all students. However, I don’t think that changing the name is the answer. Are there possible issues with the name? Sure – it’s yet another piece of jargon that can make the group seem exclusive, it isn’t intuitive what it means, it gives the impression that a teacher needs to be active in Twitter or blogging in order to join. That isn’t the biggest obstacle, though. Changing the name is less important than tweaking the culture. Mind you, I don’t have the answer to how to change the culture – I just think that that’s the question that we have to be asking right now.

If the organization was starting over, maybe it would have made sense to use the #iteachmath hashtag, but the #MTBoS is a part of the identity of this group now. A decision to change something so fundamental to the group’s identity can feel very much like a betrayal, and can seem divisive to those who have developed a tie to this hashtag that is completely tied to the blogs, the tweets, and the bloggers and tweeps.

I don’t have an answer to this, but I think I may have identified (largely by listening to very wise members of #MTBoS over the last few days, especially Anna Blinstein, Sam Shah and Peg Cagle this morning over breakfast.) the two main issues that arose in this debate.

1. How can we create an atmosphere within our community where non-members, especially elementary school math teachers and teachers of color, feel welcomed and don’t feel like outsiders?
2. How can we reach that goal while preserving the tightly knit community and the parts of its identity that are so important to members of #MTBOS?

I don’t pretend to have the answers by any means to either of these two questions. I just want to be sure that, like in math class, we are addressing the problem that is being asked, and not solving a problem that doesn’t exist.

I would ask your thoughts in the comments, but the conversation has largely been hashed out on twitter, and I suspect it will continue for a while.

# My Favorite Year End Review Activity

It’s the middle of summer, and I’m so far behind on blog posts I intended to write. All that free time in the summer seems to evaporate so quickly! It’s 2/3 of the way through July, and my first moment when I don’t have a family vacation, a daddy-daughter day, doctor’s appointments, car maintenance, work around the house, or scheduled work-related or scheduled math activities to do, so I get to share my favorite review activity. A lot of students like this too! I call it Speed Dating, and it’s fairly simple to set up. Each student is required to prepare one problem in advance. I give them the answer, but they need to work on how to solve the problem, and should make sure to ask any questions about the solution in advance if they feel unsure. If your class is large, you can break them up into smaller groups, and give each group the same set of problems. That way, too, all students working on a particular problem can come together to discuss their solutions in advance.

On the review day, each group should be set up in two circles – an inside circle and an outside circle, where each inside student is paired with an outside student. Make one larger space between two sets of desks – large enough to be able to walk through. This space will serve as the pivot point (explained later), but also makes it easier for students (and you, as the teacher) to get inside the circle. They should bring their solution with them as reference, and a place to take notes on the other solutions that they will see. I also include a small whiteboard and two different colored markers at each desk. Then, the fun starts.

I set a timer for three minutes, and the person on the inside explains their problem and solution, using the whiteboard. The person on the outside can use their own whiteboard marker to make notes or diagrams on the whiteboard if they have questions, and they can take notes on their own paper/tablet. When three minutes are up, they reverse roles, and have another three minutes for the outside student to explain their solution. After six minutes, students rotate.

All students move to the left, except for students at the pivot point, where one student in each pair wraps around, so that their partner stays on their left. Basically, you end up with a closed loop, where, given enough time, each student gets paired with every other student. During this process, each student gets to hear the solution to a wide variety of problems. In addition, every student is able to work on their explanation for their own problem, and through the extra practice, becomes a true expert in their problem, understanding it on a deeper and deeper level. Through the comments and questions they hear from their peers, they are able to focus on the trickiest parts of a problem, and refine their own solution.

When students are finished, they get time to re-write their solution and their updated solutions can be uploaded to our Google Classroom page and shared with their peers. I especially like this approach when doing a cumulative review, such as at the end of a semester in preparation for a final exam.